The list doesn't make sense for ranking in terms of "efficiency". There are trade offs with some of them in terms of accuracy vs number of function evaluations.
First of all, "Double integration" doesn't even belong on this list, as it is not a numerical integration method.
The others I would rank in terms of just plain robustness, or what I would call, meaningful accuracy:
2) trapezoidal rule
4) Higher order Newton Cotes
By the way, Romberg is a Newton Cotes type of scheme, but simply involves applying the trapezoidal rule in a very smart way to almost eliminate error without oversampling. Many times, you do numerical integration using data obtained experimentally, so using a general Newton Cotes may represent fitting noise and thereby stymying accuracy gains from the higher orders. For simplicity, nothing beats plain trapezoidal. Simpson is also very good, but a little more complicated.